Logical Reasoning:ProofProve the theorem using the basic axioms of algebra.
1. If and then ac bc c a b 0, .ac bc c and 0 Given
1 1
cac
cbcF
HG b Mult. axiom of equality
1 1a b Inverse axiom of Mult.
a b Identity axiom of Mult.
Logical Reasoning:ProofProve the theorem using the basic axioms of algebra.
2. If then a c b c a b , .
a c b c Given
a c c b c c bg bg Add. axiom of equality
a c c b c c bgbg bgbgAssoc. axiom of Add.
a b 0 0 Inverse axiom of Add.
a b Identity axiom of Add.
Logical Reasoning:ProofFind a counterexample to show that the statement is not true.
3. a b a b2 2
1 2 1 22 2 1 4 3
5 3Because , you have shown one case in which the rule is false.
5 3
Logical Reasoning:ProofFind a counterexample to show that the statement is not true.
4. a b b a
1 2 2 1
1 1
Because , you have shown one case in which the rule is false.
1 1
Logical Reasoning:ProofFind a counterexample to show that the statement is not true.
5. a b b a
1 2 2 1
05 2.
Because , you have shown one case in which the rule is false.
05 2.
Logical Reasoning:ProofUse an indirect proof to prove that the conclusion is true.
6. If , then a b b c a c
a ca b c b
Assume the conclusion is false and contradict the given statement.
Assume the opposite of is true.a c
Add. axiom of equality
a b b c Comm. axiom of Add.This statement contradicts the original statement . Therefore, it is impossible that . Therefore is true.a b b c a c
a c
Logical Reasoning:Proof
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